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randomSd -- a random homogeneous system of polynomial equations

Synopsis

Description

Generates a system of homogeneous polynomials T_i such that deg T_i = d_i. The system is normalized, so that it is on a unit sphere in the Bombieri-Weyl norm.
i1 : T = randomSd {2,3}

                               2                                      
o1 = {(.0135851 - .276691*ii)x1  + (- .0369274 - .252515*ii)x1*x2 + (-
     ------------------------------------------------------------------------
                            2                                             
     .116333 + .145332*ii)x2  + (.406186 - .00331644*ii)x1*x3 + (.392218 +
     ------------------------------------------------------------------------
                                                 2               
     .20853*ii)x2*x3 + (- .143641 - .201321*ii)x3 , (- .0235238 +
     ------------------------------------------------------------------------
                   3                           2                 
     .0507344*ii)x1  + (.264945 - .147999*ii)x1 x2 + (- .302194 +
     ------------------------------------------------------------------------
                     2                              3             
     .527688*ii)x1*x2  + (- .0121217 - .233064*ii)x2  + (.249264 +
     ------------------------------------------------------------------------
                 2                                                  
     .33538*ii)x1 x3 + (.155545 + .152282*ii)x1*x2*x3 + (- .046369 +
     ------------------------------------------------------------------------
                  2                                  2             
     .641816*ii)x2 x3 + (- .561295 - .118524*ii)x1*x3  + (.182891 +
     ------------------------------------------------------------------------
                     2                              3
     .170701*ii)x2*x3  + (- .0600091 - .252735*ii)x3 }

o1 : List
i2 : (S,solsS) = goodInitialPair T;
i3 : M = track(S,T,solsS,gamma=>0.6+0.8*ii,Software=>M2)

o3 = {{{-.0366493+.0193376*ii, .495982+.0634331*ii, .864995-.006682*ii},
     ------------------------------------------------------------------------
     #steps => 11, status  => REGULAR, last t => 1, cond#^{-1} => .556726}}

o3 : List

Ways to use randomSd :